Trees with solely \(n\)  vertices sampled proportionally to their \(w^{{\theta }}\) -weight are known in the literature as simply generated. When the sequence \({\theta }\)  is a probability measure, they correspond to Bienaymé–Galton–Watson trees with offspring distribution \({\theta }\)  conditioned on having \(n\)  vertices; similarly, \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  is the law of such a random tree conditioned to have \(n\)  vertices and \(K_{n}\)  leaves. A classical fact for trees conditioned only by their number of vertices is that an exponential tilting of \({\theta }\)  as in Lemma REF  leaves the distribution of the random trees unaffected, see e.g. the survey of Janson [1]}. Although we shall not need it, let us mention that in the context of trees conditioned both by their number of vertices and leaves, one can gain a degree of freedom by only making the exponential change to the weights \((\theta (i))_{i \geqslant 1}\)  and setting for 0 any arbitrary value.
